This blog post has to do with vertical non-permanent surfaces (VNPS), but it really is the need for more pedagogical reasoning and sense-making in our community. I don’t mean to pick on VNPS in particular–I am using it as an example only because I happen to have some data about it.

About VNPS: (I apologize for using the acronym VNPS, but “vertical non-permanent surfaces” is lot of typing.) In a nutshell, this instructional strategy involves having students working at a whiteboard/blackboard (mounted on wall or easel) or nifty erasable plastic sheets that stick to the wall instead of working at their desks. The instantiation of this instructional strategy can vary: students can be asked to work solo or in small groups; all the students might be working on the same task or different tasks; the teacher might have students stay at their stations the whole time or combine the activity with a gallery walk. There are endless variations. Search Twitter to learn more. VNPS can bring lots of wonderful benefits to a classroom, and I’m not disputing those things here. You can also find some research (here’s one example) on VNPS.

In this blog post, I would like to make the argument that VNPS makes student thinking visible in the classroom and sometimes that visibility can lower the cognitive demand of the task. This likely happened in some classrooms that I visited a few months ago.

Information about the classrooms:

- Four different teachers at the same school were implementing the same mathematical task in an Algebra 1 class and all asked students to work at whiteboards in groups of 2 or 3. The four teachers work closely together and decided on the task and implementation of the task via VNPS together.
- This task was given near the beginning of the school year, when students were beginning to develop their understanding of linear growth.
- I observed four periods on a day when the periods were about 50-minutes long. The teachers had students working at the board roughly between 15 to 30 minutes.
- I should also admit my own bias here in that I know these four teachers well and respect them highly. I look to them to gain insights about teaching mathematics.

Here is the task that students were working on.

Person A starts off with 400 widgets and gains 20 new widgets every day. Person B starts off with 50 widgets and gains 70 new widgets every day.

1) For each of the next 10 days, who will have more widgets?

2) Will there be a time when both people have the same number of widgets?

3) Is it possible for them to have the same number of widgets again past the ten day period you have explored? Explain.

Make sure that all of your answers are thoroughly explained and justified on your boards. Use any and all mathematical methods/models you can think of to explain your answers and reasoning.Overall, the quality of the student work was very high. All four teachers were careful to avoid telling students to use specific procedures (to use a graph, or a table, etc.) or to “help” students in a way that lowered the cognitive demand of the task.

(Modified version of task to protect anonymity of the teachers and students involved. The original task had a context that was much more interesting that “widgets.”)

As you might expect for this open-middle task, students used a variety of approaches for the task including tables, graphs, and symbols. Across all four of these Algebra 1 classes, I observed 39 different student groups. Out of the 39 groups, 31 of them displayed a table of numbers similar to these:

Out of the 39 groups, 11 of them displayed a graph similar to these:

Out of the 39 groups, 16 of them showed some form of algebraic expression. Eight groups set up an equation and solved for the day when the number of followers was the same. Here’s a table summarizing the frequency of various approaches that students used.

Tables were the most common approach taken up by students. That is not surprising given that it was the beginning of the year and not students might have been fluent with graphing quantities and using algebraic expressions yet. In addition, the wording of question 1 led to most students to start off by listing the number of widgets by days.

Algebraic approaches were the least common. However, Period 3 stands out in that 5 out of 8 groups used an algebraic approach that also involved solving an equation. Here is the student work for those 5 groups. (As with the other photos here, I’ve edited the photos slightly to protect the identities of the students and teachers.)

Did you notice how all five examples of student work from this class period bear a striking resemblance to each other?

- All of the student groups used “x” as the independent variable. There were other examples from other classes where the variable used was “d” (for days). Similarly, when a dependent variable was used, it was always “y” even though the objects being counted in this problem didn’t suggest the letter “y”.
- All of the student groups solved for the variable using the strategy of adding or subtracting quantities from both sides of the equations at first. And, they all showed their work in the same way. Four of the groups chose to subtract 20x as the first step.

Based on these observations and my own notes taken on what students were doing during this period, I am almost certain that students looked around the room and copied this algebraic approach to their board. One group first came up with the approach, then I observed other groups copy it from the first group and from each other. It was fascinating to watch how ideas spread throughout the room in this and other periods.

(By the way, the teacher of this Period 3 class also communicated that a significant proportion of students in that class all had the same very demanding 8th grade teacher the previous year. By all accounts, that teacher apparently had remarkably inflexible standards for how to “do math” and reinforced them using methods such as public humiliation. We wonder how much of the very similar boardwork from this period came from having had this particular approach to solving equations drilled into students the previous year, and how much a fear of public humiliation might have contributed to students copying each other.)

Evidence of students getting ideas from each other in other periods was not as strong, but I believe it also happened to some extent. For example, I observed students who saw that another group had a graph and so that led them to also include a graph. It is also interesting to me that all of the groups never had any disagreement on whether the number of widgets was 400 and 50 on day 0 or day 1. And, the fact that there were no arithmetic errors at all across all 39 work samples leads me to suspect that perhaps were checking their numbers with each other.

It seems like a pretty obvious point to make that if you’re going to have students working publicly on a task using VNPS, you might get students getting ideas or directly copying work from each other. But, I think this is an important point to make for several reasons.

First of all, the possibility of students’ revealing ideas to their peers during the enactment of a mathematical task, voluntarily or involuntarily, means that the use of VNPS has the potential to affect the cognitive demand of that task. The QUASAR project has helped our field understand how the cognitive demand of mathematical tasks tends to decrease as they are successively taken up by teachers then students.

In this case, all four teachers were very careful not to lower the cognitive demand of the task through their interactions with students during instruction. However, they might not have anticipated that the cognitive demand might have been affected by the choice of using VNPS.

Second, the possibility of students being able to see each other’s work during a mathematical task has implications for whose work gets valorized and ignored in the classroom. Imagine a hypothetical situation in which Students A, B, C are working together in a small group. And, imagine that Student X (not in their group) is a student that most others in the class perceive (accurately or not) to be mathematically competent. Student A shares a good idea with Students B and C, but they ignore her contribution and defer to Student X’s work instead, which they spy on from across the room. This hypothetical example is one possible way in which the public sharing of student work in progress might affect whose ideas get taken up.

Borrowing the language of complex instruction: VNPS could exacerbate any status issues that you might have in your classroom because interactions with any high status students are no longer restricted to the small group that those high status students belong to.

Third, the possibility of students being able to gain ideas from each other by looking around the room limits a teacher’s ability to use tasks to assess students’ abilities and understandings. Let’s return to the implementation of the widgets task described above. If the Period 3 teacher above wanted to know which solution strategies for modeling linear growth students remembered from their 8th grade math class, the VNPS enactment of this task might have led to an erroneous conclusion that many of them could proficiently work with algebraic expressions and solve equations.

I am not criticizing these four teachers for lowering the cognitive demand of the task. And in fact, there is one aspect of their use of VNPS that I think is quite brilliant: Because they did not specify whether to use tables, or graphs, or algebraic expressions, the fact that students had different approaches and then copied them to their work areas means that they had opportunities to make connections between these different representations of the scenario. And, I observed these teachers encouraging students to make these connections when they talked to individual student groups.

The purpose of highlighting these issues with VNPS is also not to demonize this very useful instructional strategy or to criticize these four teachers. Rather, I highlight these subtleties with VNPS to illustrate a more general point: when we educators talk about our teaching with each other, we tend to focus on the “hows” and “whats” rather than the “whys” and “whens”, and that incomplete transmission of ideas limits the effectiveness of those conversations.

A lot of what gets presented at conferences, written about in the Math Twitter Blog-o-Sphere, including a lot of what I have written myself, focuses on the “hows” and “whats”: how I create performance tasks when I use standards-based grading, what formulas I use in my grading scheme, what I do to handle re-dos, etc. What gets talked about less is why I use standards-based grading in the first place, when might I use that strategy over a different strategy, etc. We tend to focus on the “hows” and “whats” because that practical information is, um.., practical, so it helps us implement ideas quickly. However, the “hows” and “whats” likely don’t help us understand why something doesn’t work the way we thought it would, or how we adapt an idea to fit within our specific teaching contexts.

To effectively spread instructional innovations across a community, members of that community need to not just explain how to do X, but why they do X, when they do X instead of Y, what aspects of their teaching contexts lead them to implement X in the specific way that they have.

For example, when we talk about VNPS, let’s also talk about why we use VNPS as opposed to other instructional strategies. We shouldn’t use VNPS just because they are trendy–there should be some specific instructional purpose behind those decisions.

VNPS can be an effective strategy for getting students out of their seats and actively engaged in mathematical tasks with each other, but you could do something similar by setting up stations around the room and having manipulatives or handouts on desks as opposed to having mathematical work shared publicly around the room. What are the affordances of each in your specific context?

VNPS can be an effective way to get students to try out incomplete ideas in draft form because you can erase your work easily. However, you can achieve something similar by having personal whiteboards or shared whiteboards placed horizontally on tables. If you want student work to be shared publicly, that can still be done after the work is completed by having the personal whiteboards lifted up or placed along the wall for a gallery walk.

The specific arrangement of your classroom also matters quite a lot. One of the four teachers I mentioned above has a rather large room and even though students were working on white boards mounted around the room, they were far apart from each other that the only transmission of ideas that I observed were by nearest neighbors. In other teachers’ rooms, white boards were packed closely together and it was much easier for students to see each others’ work.

So, instructional goals, classroom culture, students’ predilections, even furniture arrangement all matter a great deal when we talk about our teaching ideas with each other. We should aim to situate our conversations about teaching innovations more thoroughly so as to enable others to make the greatest use of those ideas. If we don’t, then we increase the risk of unexpected consequences when we try out each others’ ideas and we miss out on opportunities to strengthen our own and others’ pedagogical reasoning and sense-making skills.

Think about it this way: if a student tells you the answer is 42, you will probably also ask them to justify their answer. Why then, should we as educators, only tell each other the things we’re doing in our classrooms but not also justify our ideas with pedagogical sense-making and reasoning?